# Analytic vs. formal vs. étale singularities

There are multiple definitions of when points $P$ and $Q$ on schemes $X$ and $Y$ are equisingular. One can require that either $P$ and $Q$ have isomorphic analytic (i.e. complex analytic), or formal, or étale neighborhoods. My main question is how are these three definition related. I know that for simple singularities they are equivalent, and for difficult ones I guess they are non-equivalent. Does anyone know precise statements, references, proofs, examples when these are non-equivalent notions?

The analytic and the étale case is a little bit subtler. One can ask for the adequate local rings to be isomorphic and also for isomorphic analytic/étale neighborhoods, as above. Are these two approaches equivalent?

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